[Proof of 10.24.2]
The proof is similar to the proof of Lemma 6.2 in Fleming–Soner [1]}.
Let \(\phi \in C^{1,2}_{\text{Lip}}([0,T]\times \mathbb {C};\mathbb {R})\)  and \((\bar{t},\bar{\gamma })\in [0,T]\times \mathbb {C}\)  be a strict minimizer of \(\phi -u\)  with \(\phi (\bar{t},\bar{\gamma })=u(\bar{t},\bar{\gamma })\) .
On one hand, due to that \(u_n\)  converges to \(u\)  uniformly in each compact subset of their domain, there exists a sequence \((t_n,\gamma _n)\rightarrow (\bar{t},\bar{\gamma })\)  as \(n\rightarrow \infty \) , such that \((t_n,\gamma _n)\)  is a local minimum of \(\phi -u_n\) .
On the other hand, if we set \(\tilde{\phi }(t,\gamma )\triangleq \phi (t,\gamma )+u_n(t_n,\gamma _n)-\phi (t_n,\phi _n)\) , then \((t_n,\gamma _n)\)  is a local minimum of \(\tilde{\phi }-u_n\)  and \(\tilde{\phi }(t_n,\gamma _n)=u_n(t_n,\gamma _n)\) . Hence, from the viscosity property of \(u_n\) , we have that
\(\frac{\partial }{\partial t} \tilde{\phi }(t_n, \gamma _n)+H_{n}\big (t_n, \gamma _n, \tilde{\phi }(t_n, \gamma _n), D_{x} \tilde{\phi }(t_n, \gamma _n), D_{x}^{2} \tilde{\phi }(t_n, \gamma _n)\big )\geqslant 0.\) 
